On Predicting Principal Components through Linear Mixed Models

This work introduces a Principal Component Analysis of data given by the Best Predictor of a multivariate random vector. The mixed linear model framework offers a comprehensive baseline to get a dimensionality reduction of a variety of random-effects modeled data. Alongside the suitability of using model covariates and specific covariance structures, the method allows the researcher to assess the crucial changes of a set of multivariate vectors from the observed data to the Best Predicted data. The estimation of the parameters is achieved using the extension to the multivariate case of the distribution-free Variance Least Squares method. An application to some Well-being Italian indicators shows the changeover from longitudinal data to the subject-specific best prediction by a random-effects multivariate Analysis of Variance model.